Completeness of an FO theory with a single model

Question

Prove or disprove:

If some FO theory $T$ has only one infinite model up to isomorphism (of cardinality $n$) , then T is complete.

Answer 1

This is false. Consider $L$ with only equality. Let $T= \{(\exists x)(x=x)\}$. Now, for any infinite $\kappa$ there is exactly one model up to isomorphism. However, this theory is incomplete since $T\not \models$ "There exists at least 2 elements" and $T \not \models \neg$ "There exists at least 2 elements".

Now, if you expand your assumptions and also assume $T$ has no finite models, then yes, your statement holds (this result is known as the Łoś–Vaught test).